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        Questions tagged [arithmetic-geometry]

        Diophantine equations, elliptic curves, Mordell conjecture, Arakelov theory, Iwasawa theory, Mochizuki theory.

        2
        votes
        0answers
        82 views

        A good notion of “minimal field of definition"

        Let $X$ be a variety over a separably closed field $k$. By definition of variety, there exists a subfield $k_0\subset k$ and a $k_0$-variety $X_0$ such that $$X_0\times_{k_0}k\simeq X$$ There are a ...
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        votes
        0answers
        34 views

        Uniqueness properties and faithfully flat extensions [on hold]

        Let $t : A\to B$ be a faithfully flat ring map and suppose $f,g : B\to B$ are two ring endomorphisms. Assume that there are two ring endomorphisms $f_0, g_0 : A\to A$ such that the following diagram ...
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        vote
        0answers
        22 views

        Counting geometrically irreducible components

        If we take a finite field and consider irreducible varieties over it, are there any interesting arithmetical statistics problems associated to the number of geometrically irreducible components?
        21
        votes
        1answer
        597 views

        Does anybody do $p$-adic Teichmüller theory?

        In "Foundations of $p$-adic Teichmüller theory", Mochizuki describes a theory one of whose goals (according to the author) is to generalize Fuchsian uniformization of Riemann surfaces to the $p$-adic ...
        9
        votes
        0answers
        142 views

        Grothendieck-Teichmüller conjecture and tropicalization of moduli of curves

        Abramovich, Caporaso and Payne (2014) have constructed functorial tropicalization maps from the Berkovich analytification of the moduli spaces of stable curves, $\overline{M}_{g,n}$, to the moduli ...
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        votes
        3answers
        838 views

        Tower of moduli spaces in Scholze's theory

        My question is related to another one I read here in Overflow. I am reading Scholze's papers about moduli spaces of $p$-divisible groups and elliptic curves, and I am very interested in the formal ...
        3
        votes
        0answers
        92 views

        Existence of regular hypersurface sections

        Let $X$ be a irreducible regular projective variety over $Spec(O_K)$ for some number field $K$. Is it known that there exists at least one hypersurface over $Spec(O_K)$ such that cuts $X$ in a regular ...
        6
        votes
        1answer
        127 views

        Endomorphism rings of ordinary elliptic curves

        Let's say $p$ is a prime and $t\neq 0$ is a trace of Frobenius that occurs over $\mathbb{F}_p$. The discriminant of the Frobenius polynomial is $\Delta:=t^2-4p.$ So we obtain $4p=t^2-\Delta.$ If $E$ ...
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        vote
        0answers
        192 views

        Moduli spaces of arithmetic varieties with isomorphic $l$-adic cohomology

        Given a positive integer $d$, a rational prime $l$ and a number field $K$, is it sensible to consider the moduli stack of $d$-dimensional varieties over $K$ whose $l$-adic cohomology rings are ...
        4
        votes
        0answers
        159 views

        Does etale homotopy type see the existence of rational points?

        Do there exist two smooth projective schemes over $\mathbb{Q}$ that are etale homotopy equivalent and only one of them has a $\mathbb{Q}$-point?
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        votes
        0answers
        91 views

        Computing the genus of a plane curve

        Let $b(x)=x^4 + 3x^3 + 3x^2 + 2x + 1$, and let $a(x)\in \mathbb Z[x]$ be a separable polynomial. Let $C$ be the plane curve defined by $(y^2+(x+x^2+x^3)a(x))^2-a(x)^2b(x)=0$. I would need to show that ...
        6
        votes
        1answer
        229 views

        Rigid versus log-rigid cohomology for semistable varieties

        If $K$ is a p-adic field, with maximal unramified subfield $K_0$, and $X$ is a proper semi-stable $O_K$-scheme, then there's a canonical way to make the special fibre $X_k$ into a log-scheme; and ...
        1
        vote
        0answers
        82 views

        Tate module is canonically isomorphic to a $\mathbb Z_p$-lattice on Shimura Variety of Hodge Type

        Let $(G, X)$ be a Shimura datum of Hodge type. Suppose that $K \le G(\mathbb A_f)$ is such a compact open subgroup that its $p$th component $K_p = \mathcal G(\mathbb Z_p)$ is a hyperspecial subgroup ...
        3
        votes
        0answers
        108 views

        Extension of scalars and power series rings

        Let $R$ be a ring and $R[\![x]\!]$ the formal power series ring over $R$. Suppose $R\to R’$ is either: a finite étale ring map a localization at the multiplicative set generated by one element $f\in ...
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        votes
        0answers
        117 views

        Hasse-Weil zeta function of smooth projective toric varieties

        Let $X$ be a smooth projective toric variety over a number field $K$ (assume the tori is split). As $X$ is rational, maybe the related Hasse-Weil zeta function can be well-understand, so how much do ...

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        山西福彩快乐十分钟
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